Finite Group Databasebeta

← #223234

Subgroups — #223234

Order 512.

796 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36210C211yesmaximal, characteristic
2#22797C211yesmaximal, characteristic
3#22750C211yesmaximal, characteristic
4#9907C211yesmaximal, characteristic, complemented
5#9902C211yesmaximal, characteristic, complemented
6#9136C211yesmaximal, characteristic, complemented
7#9136C211yesmaximal, characteristic, complemented
8#9128C211yesmaximal, characteristic, complemented
9#9127C211yesmaximal, characteristic, complemented
10#8728C211yesmaximal, characteristic, complemented
11#8727C211yesmaximal, characteristic, complemented
12#8590C211yesmaximal, characteristic, complemented
13#8232C211yesmaximal, characteristic, complemented
14#8046C211yesmaximal, characteristic, complemented
15#8044C211yesmaximal, characteristic, complemented
16C2×#432C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20#2859C2211yescharacteristic
21#2857C2211yescharacteristic
22#2857C2211yescharacteristic
23#2562—21no—
24#2514C412yes—
25#2509—21no—
26#2495—21no—
27#2494—21no—
28#2494—21no—
29#2485—21no—
30#2421C412yes—
31#2415C412yes—
32#2370—21no—
33#2362C412yes—
34#2347—21no—
35C4×C4⋊Q8C2211yescharacteristic
36#2285C2211yescharacteristic
37#2285C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1955—21no—
40#1924—21no—
41#1924—21no—
42#1923—21no—
43#1917—21no—
44#1869—21no—
45#1842C2211yescharacteristic
46#1831C2211yescharacteristic
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1771—21no—
52#1767C2211yescharacteristic
53#1767C2211yescharacteristic
54#1752—21no—
55C2×#333C2211yescharacteristic
56C2×#333C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C23×C8—21no—
72C2×C4⋊Q8C2311yescharacteristic
73C2×#125—21no—
74C2×C22⋊Q8C2×C412yes—
75C2×C22⋊Q8C2×C412yes—
76C2×C4×Q8C2×C412yes—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×#117—21no—
81C2×#117C2×C412yes—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#432—22no—
85#432—22no—
86#432—41no—
87#432—41no—
88#431—22no—
89#431—41no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—41no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×#97—21no—
113C2×#97—21no—
114C2×C4×C8—21no—
115#401—22no—
116#400—22no—
117#400—22no—
118#397—22no—
119#397—22no—
120#394—22no—
121#391C2×C412yes—
122#391C2×C412yes—
123#391C2×C412yes—
124#389—22no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#389—21no—
132#389—21no—
133#389C2×C412yes—
134#389C2×C412yes—
135#388—22no—
136#388—22no—
137#388—22no—
138#388—22no—
139#387—22no—
140#387—22no—
141#387—22no—
142#384C2311yescharacteristic
143#384C2311yescharacteristic
144#384C2311yescharacteristic
145#383—22no—
146#382—21no—
147#382—21no—
148#382C2×C412yes—
149#380—21no—
150#380—21no—
151#380—21no—
152#380—21no—
153#380—21no—
154#380—21no—
155#380—21no—
156#380C2×C412yes—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4C2×C412yes—
164C4×C4⋊C4C2×C411yescharacteristic
165C4×C4⋊C4D411yescharacteristic
166C4×C4⋊C4D411yescharacteristic
167C4×C22⋊C4C2×C411yescharacteristic
168C4×C22⋊C4C2×C411yescharacteristic
169C4×C22⋊C4D411yescharacteristic
170C4×C22⋊C4D411yescharacteristic
171C4×C22⋊C4D411yescharacteristic
172C4×C22⋊C4D411yescharacteristic
173C43C2311yescharacteristic
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#333—22no—
183#333—22no—
184#333—41no—
185#333—41no—
186C22×Q8C22×C412yes—
187C22×D4C22⋊C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—42no—
203C22×C8—42no—
204C22×C8—81no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209#125—42no—
210#125—42no—
211C22⋊Q8—44no—
212C22⋊Q8—82no—
213C4×Q8—82no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—41no—
217C4×D4—41no—
218C4×D4—42no—
219#117—42no—
220#117—42no—
221#117—82no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C412yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4C22⋊C411yescharacteristic
246C2×C22⋊C4C22⋊C411yescharacteristic
247C2×C22⋊C4C22⋊C411yescharacteristic
248C2×C22⋊C4C22⋊C411yescharacteristic
249C2×C22⋊C4C22⋊C411yescharacteristic
250C2×C22⋊C4C22⋊C411yescharacteristic
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2411yescharacteristic
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—42no—
290C4⋊C8—42no—
291C4⋊C8—81no—
292C4⋊C8#4112yes—
293C4⋊C8#4112yes—
294C22⋊C8—42no—
295C22⋊C8—42no—
296C22⋊C8—42no—
297C22⋊C8—42no—
298C22⋊C8—81no—
299C22⋊C8—81no—
300C22⋊C8—81no—
301#97—42no—
302#97—42no—
303#97—42no—
304#97—81no—
305C4×C8—42no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95C22⋊C411yescharacteristic
319#95C22⋊C411yescharacteristic
320#95C22⋊C411yescharacteristic
321#95C22⋊C411yescharacteristic
322#95C22⋊C411yescharacteristic
323#95C22⋊C411yescharacteristic
324C24C2×C22⋊C411yescharacteristic
325C24C2×C22⋊C411yescharacteristic
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×D4—22no—
331C2×D4—41no—
332C2×D4—41no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—84no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C22⋊C4—22no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—82no—
443C22⋊C4—82no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—41no—
465C42—41no—
466C42—44no—
467C42—82no—
468C42—82no—
469C42C2×#4111yescharacteristic
470C42C2×#4111yescharacteristic
471C42H(5,𝔽2)11yescharacteristic
472C422−1+411yescharacteristic
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23#41312yes—
487C23C22×C22⋊C411yescharacteristic
488Q8—82no—
489Q8—82no—
490D4—44no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4C4×#4111yescharacteristic
556C2×C4#52011yescharacteristic
557C2×C4#54611yescharacteristic
558C2×C4#54611yescharacteristic
559C2×C4#54711yescharacteristic
560C2×C4#54711yescharacteristic
561C2×C4#57511yescharacteristic
562C2×C4#57511yescharacteristic
563C2×C4#57511yescharacteristic
564C2×C4#57511yescharacteristic
565C2×C4#57711yescharacteristic
566C2×C4#57711yescharacteristic
567C2×C4#57811yescharacteristic
568C2×C4#57811yescharacteristic
569C8—42no—
570C8—42no—
571C8—81no—
572C8—81no—
573C8—81no—
574C8—81no—
575C8—81no—
576C8—81no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22#233911yescharacteristic
592C22C2×#41311yescharacteristic
593C22#288811yescharacteristic
594C22#289611yescharacteristic
595C22#289611yescharacteristic
596C22#289811yescharacteristic
597C22#289811yescharacteristic
598C4—21no—
599C4—21no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4#331111yescharacteristic
615C4#331311yescharacteristic
616C4#331511yescharacteristic
617C4#331711yescharacteristic
618C4#334811yescharacteristic
619C4#334911yescharacteristic
620C4#334911yescharacteristic
621C4#335011yescharacteristic
622C2—42no—
623C2—81no—
624C2#1775611yescharacteristic
625C2#1776811yescharacteristic
626C2#1776811yescharacteristic
627C2#1778011yescharacteristic
628C2#2248211yescharacteristic
629C2#2249411yescharacteristic
630C2#2249611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #9907C2semidirect
5. #9902C2semidirect
6. #9136C2semidirect
7. #9136C2semidirect
8. #9128C2semidirect
9. #9127C2semidirect
10. #8728C2semidirect
11. #8727C2semidirect
12. #8590C2semidirect
13. #8232C2semidirect
14. #8046C2semidirect
15. #8044C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#362102
#227972
#227502
#99072
#99022
#91362
#91282
#91272
#87282
#87272
#85902
#82322
#80462
#80442